Question

The sum of the digits of a two-digit number is 12. If the new number formed by reversing the digits is greater than the original number by 18, find the original number.

Answer 1

Hint: To solve this problem we must consider a number in the form of x and y and then assign them their respective place value to form a number. After this, we try to formulate some equations that are useful for solving this problem. Since, it is a two-digit problem so two variables are involved in solving the problem and hence two equations are required to obtain the possible solution.

Complete step by step answer:
An arbitrary two-digit number is represented as AB which can be expanded and rewritten as A + 10B = AB. So, by using this fact we form our number taking two variables x and y.
Now, let the unit place be x and the tenth place be y such that the number formed is xy.
As given in the problem, the sum of the digits of the number is 12.
$x+y=12\ldots (1)$
So, this is equation (1) which is helpful in solving the problem.
Also, as per the problem, if the digits are reversed then the new number formed is greater than the original by 18. So, our equation (2) would be:
$\begin{align}
  & y+10x=x+10y-18 \\
 & 9y-9x=18\ldots (2) \\
\end{align}$
Putting the value of x from equation (1) into equation (2) and solving further we get,
$\begin{align}
  & 9y-9\cdot (12-y)=18 \\
 & 9y-108+9y=18 \\
 & 18y=126 \\
 & y=7 \\
\end{align}$
Now putting the obtained value of y in the equation we get, $x=12-7=5$.
So, the obtained values of x and y are 5 and 7 respectively.
Therefore, the original number is 57.

Note: The key step for solving this problem is formulation of word problem into numerical data correctly. Once this interpretation is done, then we are left with simple mathematical calculations. Also, students must remember their assumed number and there should be no confusion about the digits taken at different place values.